Electronic communications using devices such as telephones, fax machines, mobile telephones, radios, and modems generally involve wireline electric signals and/or wireless electromagnetic signals. The study of electronic signals is called signal processing, and the study of communicating signals is called communication theory.
Electronic signals are generally characterized by at least three well-known properties—frequency, amplitude, and phase—any one of which can be used to represent information. For example, AM radio stations use electromagnetic signals that contain information in their amplitudes, and FM radio stations use electromagnetic signals that contain information in their frequencies. In general, the process of producing a signal that contains information in its frequency, amplitude, and/or phase, is called modulation, and the counterpart process of retrieving information from a such a signal is called demodulation or detection. For a radio station, for example, AM is a shorthand for amplitude modulation and FM is a shorthand for frequency modulation. In the field of communication, devices that perform modulation/demodulation have come to be known as “modems.”
AM and FM communication protocols are mainly used in radios that communicate analog audio information, and other communication protocols are generally used to communicate digital information. One popular digital communication protocol is known as “I/Q modulation,” in which the communication signal is a combination of two amplitude-modulated sinusoidal signals that have the same frequency but that are π/2 radians apart in phase, i.e., in “quadrature.” Thus, another name for I/Q modulation is quadrature amplitude modulation, or QAM. The “I” in I/Q modulation refers to information associated with the one of the signals (e.g., cosine), and the “Q” refers to information associated with the other signal (e.g., sine).
A fundamental concept that is used with signal processing and communication theory, and especially with I/Q modulation, is the representation of electronic signals as complex functions, i.e., functions that have a real and an imaginary part. Complex functions can be plotted on a complex plane in which the real part of the function corresponds to values on the horizontal axis and the complex part of the function corresponds to values on the vertical axis. The use of complex functions is a conceptual tool that provides a convenient way to represent signal amplitude and phase. For example, suppose a signal is characterized by the complex function s(t)=r(t)+j·m(t), where r(t) and m(t) are real-valued functions and j designates the imaginary part of the complex function. Using the complex function, the magnitude of the signal s(t) can be computed by |s(t)|=√{square root over ((r(t))2+(m(t))2)}{square root over ((r(t))2+(m(t))2)}, and the phase of the signal can be computed by
      ∠s    ⁡          (      t      )        =            arctan      ⁡              (                              m            ⁡                          (              t              )                                            r            ⁡                          (              t              )                                      )              .  As another example, operations that change the frequency content of a signal can also be described using complex numbers and Fourier transforms. For example, one skilled in the art will recognize that multiplying a signal s(t) with the complex sinusoid ejωct=cos(ωct)+j·sin(ωct) in the time domain will produce a resulting signal s′(t)=s(t)ejωct in which the frequency content of s′(t) is shifted by ωc compared to s(t).
In I/O modulation, the transmitted signal ideally contains two signals that are in quadrature, such as cosine and sine signals, each having a particular amplitude. For a theoretical explanation, suppose the transmitted signal iss(t)=sl,kδ(t−kT)cos(ωct)+sQ,kδ(t−kT)sin(ωct),where ωc is the angular frequency,
      T    =                  2        ⁢        π                    ω        c              ,k is an integer, δ(t−kT) is the time-shifted ideal impulse function, and sl,k and sQ,k are real values. Further suppose that the values of sl,k and sQ,k are chosen to satisfy the equations sl,k=Akcos(θk) and sQ,k=Aksin(θk). Then, the transmitted signal becomess(t)=Akcos(θk)δ(t−kT)cos(ωct)+Aksin(θk)δ(t−kT)sin(ωct).Recognizing that cos(a−b)=cos(a)cos(b)+sin(a)sin(b), the transmitted signal can be expressed ass(t)=δ(k−kT)Akcos(ωct−θk),which shows that particular choices of values for sI,k and sQ,k can result in different amplitude and phase for the transmission signal. Accordingly, I/Q modulation can represent information by modulating the amplitude and phase of the transmission signal. It is suitable to note here that the impulse function is purely theoretical and, in reality, can be replaced by some pulse-shaped function p(t).
In the field of digital communication, the set of all available values sl,k and sQ,k is called a “constellation,” and each available pair of values (sl,k, sQ,k) is called a signal point. When the constellation is designed such that Ak does not change for different signal points and only θk changes, the modulation is called phase-shift keying, or PSK. When the constellation is designed such that θk does not change for different signal points and only Ak changes, the modulation is called amplitude-shift keying, or ASK. In general, however, both Ak and θk may change. Also, the number M of signal points in a constellation determines the amount of information that is associated with each signal point. In general, each signal point can represent b=└log2M┘ bits of information, and usually M is a power of two.
Ultimately, the desired operation of a transmitter is to perform frequency, amplitude, and/or phase modulation, and circuits exist that can do so directly. For example, a voltage-controlled oscillator can be used to perform frequency modulation. If frequency modulation is not needed, however, it may be simpler to use I/Q modulation. For example, I/Q modulation can require generating two quadrature signals, varying the amplitudes of the signals according to a constellation, and then combining the amplitude-modulated signals to, effectively, produce an amplitude and/or phase modulated transmission signal. Additionally, I/Q modulation may also simplify the detection/demodulation of the transmission signal. Detection of an I/Q modulated signal will be described later herein.
One complication that can arise in I/Q modulation is that the quadrature oscillators of a transmitter and/or receiver may not be ideal and may actually not be in quadrature, giving rise to phase mismatch. Another complication is that the quadrature oscillators may provide periodic signals that have different amplitudes, giving rise to amplitude mismatch. These complications will be referred to herein as “I/Q mismatch.” I/Q mismatch can introduce unwanted frequency components into a signal and can degrade the performance of a communication system. While the “I” and “Q” oscillators in a transmitter may be carefully designed and manufactured, I/Q mismatch can nevertheless occur because of temperature dependencies and/or other phenomena affecting oscillators. Therefore, there is continued interest in providing communication systems that can operate effectively when I/Q mismatch occurs.